Position heaps are index structures of text strings used for the string matching problem. They are rooted trees whose edges and nodes are labeled and numbered, respectively. This paper is concerned with variants of the inverse problem of position heap construction and gives linear-time algorithms for those problems. The basic problem is to restore a text string from a rooted tree with labeled edges and numbered nodes. In the variant problems, the input trees may miss edge labels or node numbers which we must restore as well.